Question

A perfectly elastic particle is projected with a velocity v on a vertical plane through the line of greatest slope of an inclined plane of elevation α.If after striking the plane, the particle rebounds vertically show that it will return to the point of projection at the end of time equal to?

Answer 1

$\huge\underline{{\green{Aɴsᴡᴇʀ:-}}}$

Let angle of projection of the particle with the plane is $\sf \theta$ .

And it's displacement along y-axis becomes zero in time t.

$\mapsto \sf y = u_y t+1/2 a_y t^2 \\\\\\ \implies \sf t = 2v sin\theta /gcos\alpha....(1)$

$\\ \\ \\ \mapsto \: \sf \: vsin \alpha = vcos \theta - gsin \theta \: t...(2) \\ \\ \\ \implies \sf \: vcos \alpha = vsin \theta - gcos \theta \: t...(3)$

From eqn 2 :

$\\ \\ \\ \implies \sf \: vsin \alpha = vcos \theta \: - gsin \alpha \times \frac{2vsin \theta}{gcos \alpha} \\ \\ \\ \implies \sf \: v = \frac{vcos \theta}{sin \alpha } - \frac{2vsin \theta}{cos \alpha } ....(4)$

Now, substituting value of 't' in eqn (1)&(2) :

$\\ \\ \\ \implies \sf \: vcos \alpha = vsin \theta \: - gcos \alpha ( \frac{2vsin \theta}{gcos \alpha } )$

Now ,

$\\ \\ \\ \implies \sf \: tan \alpha = \frac{cos \theta.2sin \theta.tan \alpha }{sin \theta} \\ \\ \\ \implies \sf \: cot \theta - 2tan \alpha \\ \\ \\ \therefore \sf cos \theta = \frac{3tan \alpha }{ \sqrt{1 + 9 {tan}^{2} \alpha} }$

Total time taken by particle is equal to the sum of time taken from O to P and P to Q and from Q to P to Q.

Thus ,

» total time = 2t + 2t' = 2(t+t')

» We have t = v/g

$\\ \\ \\ \implies \sf \: total \: time = 2( \frac{2vsin \theta}{gcos \alpha } + \frac{v}{g} ) \\ \\ \\ \implies \sf \: 2( \frac{2vsin \theta}{gos \alpha } + \frac{1}{g} ( \frac{vcos \theta}{sin \alpha } - \frac{2vsin \theta}{cos \alpha } ) \\ \\ \\ \implies \sf \: \frac{2vcos \theta}{gsin \alpha } = \frac{2v( \frac{3tan \alpha }{ \sqrt{1 + 9 {tan}^{2} \alpha} } )}{gsin \alpha }$

After solving we get :

$\bigstar \orange {\boxed{ \sf \: total \: time = \frac{6v}{g \sqrt{1 + 8 {sin}^{2} \alpha } } }}$

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Answer 2

Explanation:

yeahh idk but the correct answer is correct "deleted account".